Reformulation and Generalization Of Van Aubel's Theorem Using Complex Variables

This research aims to reformulate and generalize Van Aubel’s Theorem using a complex number approach. Classically, Van Aubel’s Theorem states that if squares are constructed externally on the sides of an arbitrary quadrilateral, then the line segments connecting the centers of opposite squares are congruent and perpendicular to each other. Through a complex algebra approach, this research systematically reconstructs the proof of the theorem and extends its application to more complex geometric forms. The representation of points, rotations, and lines in the complex plane is employed to reprove and develop generalizations of the theorem. The results demonstrate that the use of complex numbers facilitates geometric analysis and reveals new patterns in geometric relationships. This research contributes to the advancement of modern geometry through algebraic methods and opens opportunities for further exploration of complex numbers in solving other geometric problems.